# questions on Deligne's letter to Piatetski-Shapiro

Now, that the letter is to be found here: http://www.math.ias.edu/~jaredw/DeligneLetterToPiatetskiShapiro.pdf ,I'd have a couple of questions concerning it. Actually, I have a problem only with the very last page. Could any one explain with more details the reasoning after Deligne proves $$H^0 = \oplus _{\chi} Hom _{H(\mathbb{R}) \times H(\mathbb{Q} _p)} (Sym ^k (V) \otimes ..., L_0)$$

1) Is the representation of $H(\mathbb{A} ^f)$ he talks about, the one which comes from the action of $H(\mathbb{A} ^f)$ on $L_0 (...)$ inside the $Hom$?

2) Why "the following representation occurs in $\kappa (\mu)$"?

3) Why "supercuspidal representations cannot occur outside" $\kappa (\mu)$ and what are references to which Pierre Deligne alludes (especially I am interested in "Langlands + vanishing cycle" method)?

4) How he deduces Theorem C from all this?

I hope, it is not too much to ask in one post.

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Have you looked at Carayol's Ann Sci Ec Norm Sup paper from 1983 or so, where he does everything in slightly more generality and adds in a lot of careful detail? In particular all the details of the vanishing cycle computation are there. – Kevin Buzzard Feb 25 '11 at 11:13
Yes, I've read "Sur les representations l-adiques..." but it seems like I will have to study it more, especially chapters 6 and 10 of that paper, where I think, Carayol does the same thing as Deligne but in Shimura curve context. Nevertheless, I would still be very grateful if someone more knowledgable would explain the above piece of Deligne's letter with more details. Hopefully, it would be useful not only for me. – Przemyslaw Chojecki Feb 25 '11 at 13:06